
Direct measurement attack Before of after the QPA process is completed, \(P_i\)  can measure the register \(h,t\)  or \(e_i\)  they own to obtain any useful information;

Pre-period-finding attack Before the QPA process is completed, \(P_i\)  apply \(\mathcal {QFT}^\dagger \)  to his own register \(h\)  or \(t\)  to obtain the LCM of the parties who have completed their operations.

Post-period-finding attack \(P_i\)  copies a \( \left|j\right\rangle \)  (using \(\mathcal {CNOT}\)  gate) when he get the register \(t\) . After the QPA process is completed and before step 5, \(P_i\)  applies \(\mathcal {QFT}\)  on his copy to obtain any useful information (the reason we choose \(\mathcal {QFT}\)  is that \( \left|\widehat{f}(l)\right\rangle =\frac{1}{\sqrt{T}}\sum _{k\in \left[ T \right]}e^{-\frac{\imath 2\pi lk}{T}} \left|f(k)\right\rangle \)  is from \( \left|f(l)\right\rangle \)  by \(\mathcal {QFT}^\dagger \)  [1]}).

